Integral oddity

[lrdheat]

Integral 1/(x^2 * sqrt(x^2 + 4)) should be (-sqrt(x^2 + 4))/4*x

The prime gives (-x -sqrt(x^2 + 4))/4*x after factorizing or after simplifying. Is this a bug?

[lrdheat]

Version 2016 04 14 (10077)

[lrdheat]

When integrating over a specific range such as pi/6 to pi/4, the exact answer converts to a correct approximate answer of ~.303

Am I missing something?

[parisse]

(07-31-2016 09:06 PM)lrdheat Wrote:  Integral 1/(x^2 * sqrt(x^2 + 4)) should be (-sqrt(x^2 + 4))/4*x

The prime gives (-x -sqrt(x^2 + 4))/4*x after factorizing or after simplifying. Is this a bug?

No, because antiderivatives are defined up to a constant.

[lrdheat]

I must be math challenged/rusty...is not the "-x" that Prime produced a variable? How are the 2 answers equivalent?

What has me befuddled is that when made into a definite integral, Prime reports a correct answer.

[Dieter]

(08-01-2016 10:08 PM)lrdheat Wrote:  I must be math challenged/rusty...is not the "-x" that Prime produced a variable? How are the 2 answers equivalent?

If you simplify the Prime result...

Code:

   –x – sqrt(x²+4)
   ---------------
         4x

   –x     sqrt(x²+4)
=  --  –  ----------
   4x         4x

            sqrt(x²+4)
=  –1/4  –  ----------
                4x

            sqrt(x²+4)
= const  –  ----------
                4x

...you get the same antiderivative plus a constant.

Dieter

[lrdheat]

Thanks!

For some reason, I was seeing the denominator as "4" instead of "4*x".